3.129 \(\int \frac {1}{\sqrt {2+5 x^2-5 x^4}} \, dx\)

Optimal. Leaf size=48 \[ \sqrt {\frac {2}{\sqrt {65}-5}} F\left (\sin ^{-1}\left (\sqrt {\frac {10}{5+\sqrt {65}}} x\right )|\frac {1}{4} \left (-9-\sqrt {65}\right )\right ) \]

[Out]

EllipticF(x*10^(1/2)/(5+65^(1/2))^(1/2),1/4*I*10^(1/2)+1/4*I*26^(1/2))*2^(1/2)/(-5+65^(1/2))^(1/2)

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Rubi [A]  time = 0.10, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1095, 419} \[ \sqrt {\frac {2}{\sqrt {65}-5}} F\left (\sin ^{-1}\left (\sqrt {\frac {10}{5+\sqrt {65}}} x\right )|\frac {1}{4} \left (-9-\sqrt {65}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[2 + 5*x^2 - 5*x^4],x]

[Out]

Sqrt[2/(-5 + Sqrt[65])]*EllipticF[ArcSin[Sqrt[10/(5 + Sqrt[65])]*x], (-9 - Sqrt[65])/4]

Rule 419

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(1*EllipticF[ArcSin[Rt[-(d/c),
2]*x], (b*c)/(a*d)])/(Sqrt[a]*Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] &
& GtQ[a, 0] &&  !(NegQ[b/a] && SimplerSqrtQ[-(b/a), -(d/c)])

Rule 1095

Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[2*Sqrt[-c], I
nt[1/(Sqrt[b + q + 2*c*x^2]*Sqrt[-b + q - 2*c*x^2]), x], x]] /; FreeQ[{a, b, c}, x] && GtQ[b^2 - 4*a*c, 0] &&
LtQ[c, 0]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {2+5 x^2-5 x^4}} \, dx &=\left (2 \sqrt {5}\right ) \int \frac {1}{\sqrt {5+\sqrt {65}-10 x^2} \sqrt {-5+\sqrt {65}+10 x^2}} \, dx\\ &=\sqrt {\frac {2}{-5+\sqrt {65}}} F\left (\sin ^{-1}\left (\sqrt {\frac {10}{5+\sqrt {65}}} x\right )|\frac {1}{4} \left (-9-\sqrt {65}\right )\right )\\ \end {align*}

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Mathematica [C]  time = 0.06, size = 52, normalized size = 1.08 \[ -i \sqrt {\frac {2}{5+\sqrt {65}}} F\left (i \sinh ^{-1}\left (\frac {1}{2} \sqrt {5+\sqrt {65}} x\right )|\frac {1}{4} \left (-9+\sqrt {65}\right )\right ) \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/Sqrt[2 + 5*x^2 - 5*x^4],x]

[Out]

(-I)*Sqrt[2/(5 + Sqrt[65])]*EllipticF[I*ArcSinh[(Sqrt[5 + Sqrt[65]]*x)/2], (-9 + Sqrt[65])/4]

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fricas [F]  time = 0.76, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-5 \, x^{4} + 5 \, x^{2} + 2}}{5 \, x^{4} - 5 \, x^{2} - 2}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-5*x^4+5*x^2+2)^(1/2),x, algorithm="fricas")

[Out]

integral(-sqrt(-5*x^4 + 5*x^2 + 2)/(5*x^4 - 5*x^2 - 2), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {-5 \, x^{4} + 5 \, x^{2} + 2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-5*x^4+5*x^2+2)^(1/2),x, algorithm="giac")

[Out]

integrate(1/sqrt(-5*x^4 + 5*x^2 + 2), x)

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maple [B]  time = 0.11, size = 80, normalized size = 1.67 \[ \frac {2 \sqrt {-\left (-\frac {5}{4}+\frac {\sqrt {65}}{4}\right ) x^{2}+1}\, \sqrt {-\left (-\frac {5}{4}-\frac {\sqrt {65}}{4}\right ) x^{2}+1}\, \EllipticF \left (\frac {\sqrt {-5+\sqrt {65}}\, x}{2}, \frac {i \sqrt {10}}{4}+\frac {i \sqrt {26}}{4}\right )}{\sqrt {-5+\sqrt {65}}\, \sqrt {-5 x^{4}+5 x^{2}+2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-5*x^4+5*x^2+2)^(1/2),x)

[Out]

2/(-5+65^(1/2))^(1/2)*(1-(-5/4+1/4*65^(1/2))*x^2)^(1/2)*(1-(-5/4-1/4*65^(1/2))*x^2)^(1/2)/(-5*x^4+5*x^2+2)^(1/
2)*EllipticF(1/2*x*(-5+65^(1/2))^(1/2),1/4*I*10^(1/2)+1/4*I*26^(1/2))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {-5 \, x^{4} + 5 \, x^{2} + 2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-5*x^4+5*x^2+2)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/sqrt(-5*x^4 + 5*x^2 + 2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{\sqrt {-5\,x^4+5\,x^2+2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(5*x^2 - 5*x^4 + 2)^(1/2),x)

[Out]

int(1/(5*x^2 - 5*x^4 + 2)^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {- 5 x^{4} + 5 x^{2} + 2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-5*x**4+5*x**2+2)**(1/2),x)

[Out]

Integral(1/sqrt(-5*x**4 + 5*x**2 + 2), x)

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